What is the area of the parabola bounded by the latus rectum?
Direction: Consider the following for the two (02) items that follow : The area bounded by the parabola y^{2}=kx and the line x=k, where k \gt 0, is \frac{4}{3} square units.
- A. 1/6 square unit ✓
- B. 2/3 square unit
- C. 1 square unit
- D. 4/3 square units
Correct Answer: A. 1/6 square unit
Explanation
We determined k=1, so the parabola is y^2 = x. The equation of the standard parabola is y^2 = 4ax, matching this gives 4a = 1 \implies a = 1/4. The area bounded by the latus rectum (the line x=a=1/4) is 2\int_{0}^{1/4} \sqrt{x} \,dx = 2\left[\frac{2}{3}x^{3/2}\right]_0^{1/4} = \frac{4}{3} \left(\frac{1}{4}\right)^{3/2} = \frac{4}{3}\left(\frac{1}{8}\right) = \frac{1}{6} square units.
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